

A002106


Number of transitive permutation groups of degree n.
(Formerly M1316 N0504)


16



1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117, 164, 59, 7, 25000, 211, 96, 2392, 1854, 8, 5712, 12, 2801324, 162, 115, 407, 121279, 11, 76, 306, 315842, 10, 9491, 10, 2113, 10923, 56, 6
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OFFSET

1,3


COMMENTS

It is conjectured that this is the number of Galois groups for irreducible polynomials of order n. (All such Galois groups are transitive.)  Charles R Greathouse IV, May 28 2014


REFERENCES

G. Butler and J. McKay, personal communication.
C. C. Sims, Computational methods in the study of permutation groups, pp. 169183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..47.
G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863911.
G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863911. [Annotated scanned copy]
John J. Cannon and Derek F. Hol, The transitive permutation groups of degree 32
F. N. Cole, Note on the substitution groups of six, seven, and eight letters, Bull. Amer. Math. Soc. 2 (1893), 184190. Gives a(8)=48 instead of 50.
Computational Algebra Group, Summary of New Features in Magma V2.21
J. Conway, A. Hulpke, and J. McKay, On Transitive Permutation Groups, LMS Journal of Computation and Mathematics 1 (1998), pp. 18. See especially Appendix A.
D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 3337. [Annotated copy]
Derek Holt and Gordon Royle, A Census of Small Transitive Groups and VertexTransitive Graphs, arXiv:1811.09015 [math.CO], 2018.
A. Hulpke, Transitive groups of small degree
A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 130.
E. G. Köhler, F. H. Lutz, Triangulated manifolds with few vertices: Vertextransitive triangulations, arXiv:math/0506520 [math.GT], 2005.
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269284. See page 280.
G. A. Miller, On the lists of all the substitution groups that can be formed with a given number of elements, Bull. Amer. Math. Soc., 2 (1896), 138145.
Wikipedia, Inverse Galois problem
Index entries for sequences related to groups
Index entries for "core" sequences


EXAMPLE

a(3)=2: A_3 and S_3.


PROG

(GAP) a:=function(n)
return Length(AllTransitiveGroups(NrMovedPoints, n));
end; # Charles R Greathouse IV, May 28 2014


CROSSREFS

Cf. A000001, A000019, A177244, A186277.
Sequence in context: A154698 A063786 A121304 * A232316 A184604 A064630
Adjacent sequences: A002103 A002104 A002105 * A002107 A002108 A002109


KEYWORD

nonn,core,hard,more,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Corrected and extended to degree 31 by Alexander Hulpke (Alexander.Hulpke(AT)Math.RWTHAachen.DE), Aug 15 1996
Further corrections from Alexander Hulpke, Feb 19 2002
Degree 32 extended by Artur Jasinski, Feb 17 2011
Extended to degree 47 by Gabriel Verret, May 07 2016


STATUS

approved



